A senior mathematician who will remain nameless recently said in a talk, "there is nothing left to prove". In context, he was referring to the possibility that we are running out of math problems. People who heard laughed, and first-year calculus students might disagree. Was it said as a joke?
Because of the infinite nature of mathematics, there will always be new problems. On the other hand, there are only finitely many theorems we'll ever know; only finitely many that we'll ever be interested in. Are we close to knowing all the interesting theorems? Is the increasing specialisation of the literature a sign of a future with a thousand subfields each with only one or two devotees?
Truthfully, I don't think math is running out of problems at all. I think it's more like good, nonspecialist exposition isn't really keeping up with the rapid development of mathematics and so we know less and less about what our colleagues are doing. So we should attempt to prevent the future where every person is their own research field. Here are some ways we could do that:
- Make part of your introduction in your paper understandable to a much wider range of mathematicians. This will encourage more collaboration and cross-disciplinary understanding. For example, once I was actually told by a journal to cut out a couple of pages from a paper because it was well-known to (probably ten) experts, even though that material was literally not written down anywhere else! Journals should actually encourage good exposition and not a wall of definition-theorem-proof.
- Have the first twenty minutes of your talk understandable by undergraduates. Because frankly, this is the only way mathematicians (especially young ones) in other fields will actually understand the motivation of your work. How are we supposed to ask good questions when we can't figure out where our research fits in with the research of others?
- Use new avenues of mathematical exposition like blogs and nontechnical articles. Other fields like physics and biology appear in magazines like Scientific American and have an army of people working to make specialised work understandable to the nonspecialist.
- Encourage new, simplified proofs or explanations of existing results. And by 'encourage', I mean count high-quality, expository papers on the level of original results in determining things like tenure and jobs! There are already journals that publish these types of papers. Chances are, any expository paper will actually help at least as many people as an original result, perhaps more. And there are still hundreds of important papers that are very difficult if not impossible to read (even by many experts), with no superior alternative exposition available.
I think it's been a long-lived fashion in mathematics to hide the easy stuff in favour of appearing slick ever since one dude tried to hide how he solved the cubic from another dude, and it's probably something we can give up now.